Formulate how many plies are the DTMs if there are multiple checkmate variations (Depth to Mate ignoring 50-move rule) for a random N-piece chess position if any checkmate exists, (and also consider the positions that there is only one checkmate variation) (N is at most 32). Consider the condition that one checkmates the other, not draw. (without neglecting ONLY THE MANDATORY FIDE draw rules and the DEAD POSITION rule). Assume that losing side tries to maximize DTM and winning side tries to minimize DTM (Losing side fighting to delay checkmate, winning side trying to deliver the checkmate as quickly as possible) such that both sides play optimally starting from the position.



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